148
ANALYTICAL MECHANICS
ILLUSTRATIVE EXAMPLES.
1. Find the center of mass of an octant of a homogeneous sphere, (a) Suppose the bounding surfaces to be
x2 + 2/2 + z2 = a2, 3 = 0, y = 0, and z = 0. Then the limits of integration are
x = 0 and x = a,
2/ = 0 and y = \/a2 — #2,
25 = 0 and z = Va2 — x2 — y2.
Therefore
nVa2-a-2 Va2-a;2-y2 I xdxdy dz y. -_ - ______JQ___________
nC dx dy dz J o
and by symmetry y = z = —•-
o
(b) Suppose the equations of the bounding surfaces to be given in spherical coordinates, then we have
r = a, 0 = , 0 = 0, and 0 = -
The limits of integration are
r = 0 and r = a, 0 = 0 and 0 = ^,
and 0 = ~ -
FIG. 81.
,.,„.„...._ xiiciciuic
fl TT
j^2 j^2 j^ar3sin2 0 cos0dr d6 dcf>
_ . .. .
[a; = r sin ^ cos0] .
3a
; 8 *